Taylor - Partial Differential Equations II 2ed Spr

文件名称: Taylor - Partial Differential Equations II 2ed Springer 2011.pdf.pdf

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详细说明：Taylor - Partial Differential Equations II 2ed Springer 2011 Taylor - Partial Differential Equations II 2ed Springer 2011 Michael E. Taylor Partial Differential Equations Il Qualitative Studies of Linear Equations Second edition ②$p ringer Michael E. Taylor Department of Mathematics University of North Carolina Chapel Hill, NC 27599 USA met math. unc. edu ISSN0066-5452 ISBN978-1-44197051-0 e-ISBN978-1-4419-7052-7 DOⅠ10.1007/978-1-44197052-7 Springer New York dordrecht Heidelberg London Library of Congress Control Number: 2010937758 Mathematics Subject Classification(2010): 35AO1, 35A02, 35J05, 35J25, 35K05, 35L05, 35Q30 35Q35,35S05 C Springer Science+Business Media, LLC 1996, 201 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher(Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights Printed on acid-free paper SPringerispartofSpringerScience+businessMedia(www.springer.com) To my wife and daughter, Jane hawkins and Diane taylor Contents Contents of volumes i and Ill Preface,........ xiii 7 Pseudodifferential operators I The Fourier integral representation and symbol classes 2 Schwartz kernels of pseudodifferential operators.......... 3 Adjoints and products 4 Elliptic operators and parametrices 25058 5L-estimates 6 Gardings inequality 22 7 Hyperbolic evolution equations 8 egorov’ s theorem 26 9 Microlocal regularity.….… 0 10 Operators on manifolds 33 11 The method of layer potential 36 12 Parametrix for regular elliptic boundary problems......... 47 13 Parametrix for the heat 56 14 The Weyl calculus 67 15 Operators of harmonic oscillator type................ 80 Ref 88 8 Spectral Theory 91 I The spectral theorem 92 2 Self-adjoint differential operato 3 Heat asymptotics and eigenvalue asymptotics. ......................106 4 The Laplace operator on Sn 113 5 The Laplace operator on hyperbolic space.............. 123 6 The harmonic oscillator 126 7 The quantum Coulomb problem 135 8 The Laplace operator on cones .149 References ,171 9 Scattering by Obstacles......................... 175 1 The scattering problem........................177 2 Eigenfunction expansions 186 3 The scattering oper viii contents 4 Connections with the wave equation 197 6 Translation representations and the lax- Phillips…………205 5 Wave operators semigroup z(t) 7 Integral equations and scattering poles 8 Trace formulas; the scattering phase..…………,23 9 Scattering by a sphere................. 239 10 Inverse problems I 248 11 Inverse problems II 12 Scattering by rough obstacles 266 a Lidskii's trace theorem ∴...275 R eferences ...277 10 Dirac Operators and Index Theory 281 1 Operators of Dirac type........................ 283 2 Clifford algebras 289 3 Spinors 294 4 Weitzenbock formulas 300 5 Index of Dirac operators 306 6 Proof of the local index formula 309 7 The Chern-Gauss-Bonnet theorem ..................................316 8 Spin m 320 9 The Riemann-Roch theorem 325 10 Direct attack in 2-D 11 Index of operators of harmonic oscillator type ........... 345 Refe 58 11 Brownian Motion and Potential Theory.................361 1 Brownian motion and wiener measure 363 2 The Feynman-Kac formula...................... 370 3 The dirichlet problem and diffusion on domains with boundary..375 4 Martingales, stopping times, and the strong Markov property 384 5 First exit time and the poisson integral ..394 6 Newtonian capacity 398 7 Stochastic integrals 412 8 Stochastic integrals, II 423 9 Stochastic differential equations 430 10 Application to equations of diffusion. ................................437 a The T product formula 448 References ..454 12 The a-Neumann Problem .457 A Elliptic complexes 460 The a-complex 465 2 Morrey's inequality, the Levi form, and strong pseudoconvexity. 469 3Thne2- estimate and some consequences……………472 Contents 4 Higher-order subelliptic estimates 476 5 Regularity via elliptic regularization.………480 6 The Hodge decomposition and the d-equation. .....................483 7 The Bergman projection and Toeplitz operators 487 8Thea- Neumann problem on(0,q)- forms.…………………494 9 Reduction to pseudodifferential equations on the boundary.... 503 10 The d-equation on complex manifolds and almost complex manifolds............... ...516 B Complements on the Levi form 27 C The Neumann operator for the dirichlet problem 531 References .............................................................53.5 C Connections and Curvature....................... 539 I Covariant derivatives and curvature on general vector bundles... 540 2 Second covariant derivatives and covariant-exterior derivatives.. 546 3 The curvature tensor of a riemannian manifold 548 4 Geometry of submanifolds and subbundles ..560 5 The gauss-Bonnet theorem for surfaces 574 6 The principal bundle picture 586 7 The chern-Weil construction 594 8 The Chern-Gauss-Bonnet theorem . .................................598 References . .............608 Index 611

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